listen with half an ear — not giving your full attention to what is being said or played.
I couldn’t find any history or stories about this phrase, which I guess is not surprising, since the meaning is pretty obvious. However, I did find a whole long list of idioms about the ear, though no history about these either. Here are some of them:
be all ears — very eager to hear; acutely attentive
Pretty much the opposite of the first one! In some cases, the phrase is intended to convey that the listener is happy to pay attention. In other cases, it may reflect a curiosity about gossip. Finally, it can be an expression of frustration by the listener, who has encountered a difficult situation or problem for which he has not found a solution. As such, he is very open and interested in listening to someone who may be able to help him achieve a resolution.
bend somebody’s ear — to talk endlessly to someone, often about a problem that you have
be out on your ear — to be forced to leave (a job, etc.) because you have done something wrong, or because your work is not good enough
be up to your ears — to have a lot to deal with, to be very busy or have more of something than you can deal with
cock an ear — to listen carefully and with a lot of attention
easy on the ear — pleasant to listen to
fall on deaf ears — to be ignored or not noticed by other people
have something coming out of your ears — to have a lot of something, especially more than you need
have somebody’s ear — get someone’s favorable attention
keep your ears open — to be quick to notice or hear things
keep your ear to the ground — make sure that you always find out about the most recent developments in a particular situation or pay attention to everything that is happening around you and to what people are saying. It is figurative, based on the story that trackers could hear approaching horses and so on, by lying with their ear on the ground.
There, have you heard enough?